Stochastic processes as a research area focuses on the mathematical modeling and analysis of systems that evolve randomly over time, typically formalized as families of random variables indexed by time or space. It encompasses foundational theory for Markov processes, martingales, Lévy processes, Gaussian processes, and point processes, with emphasis on properties such as stationarity, ergodicity, and mixing. Research addresses limit theorems, stochastic calculus, stochastic differential equations, and probabilistic potential theory. Applications span quantitative finance, statistical physics, queuing theory, population dynamics, signal processing, and machine learning, where stochastic processes provide rigorous frameworks for uncertainty quantification and temporal or spatial dependence.
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